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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Dispersive body waves ： ウィキペディア英語版 Dispersive body waves Dispersive body waves is an important aspect of seismic theory. When a wave propagates through subsurface materials both energy dissipation and velocity dispersion takes place. Energy dissipation is frequency dependent and causes decreased resolution of the seismic images when recorded in seismic prospecting. The attendant dispersion is a necessary consequence of the energy dissipation and causes the high frequency waves to travel faster than the low-frequency waves. The consequence for the seismic image is a frequency dependent time-shift of the data, and so correct timings for lithological identification cannot be obtained. ==Basics== When we know the energy dissipation (attenuation), we can calculate the time shift due to dispersion because there is a relation between attenuation and the dispersion in a seismic media.Dispersion equations are obtained from the application of an integral transform in the frequency domain that are of the Kramers-Krönig type. This effect is described in the article ‘Dispersive body waves’ by Futterman (1962).〔Futterman (1962) ‘Dispersive body waves’. Journal of Geophysical Research 67. p.5279-91〕 For a better understanding of dispersion waves in seismic medias I would recommend the book 'Seismic inverse Q-filtering' by Yanghua Wang (2008). He discuss the theory of Futterman and starts with the wave equation:〔Wang 2008, p. 60〕 :$\backslash frac\; -\; ikU(r,w)=0\; \backslash quad\; (1.1)$ where U(r,w) is the plane wave of radial frequency w at travel distance r, k is the wavenumber and i is the imaginary unit. Reflection seismograms record the reflection wave along the propagation path r from the source to reflector and back to the surface. Equation (1.1) has an analytical solution given by :$U(r+\backslash bigtriangleup\; r,w)\; =U(r,w)\backslash exp\; (ik\backslash bigtriangleup\; r)\; \backslash quad\; (1.2)$ Where k is the wave number. When the wave propagates in inhomogeneous seismic media the propagation constant k must be a complex value that includes not only an imaginary part, the frequency-dependent attenuation coefficient, but also a real part, the dispersive wavenumber. We can call this K(w) a propagation constant in line with Futterman.〔Futterman (1962) p.5280〕 :$K(iw)\; =k(w)+\; i\; a(w)\; \backslash quad\; (1.3)$ Here a(w) is taken positive to assure that energy is lost from the wave to the medium. Now when K(w) are separated into a real part for attenuation and an imaginary part for dispersion, we can introduce the Kramer-Krönig relation by a Hilbert transform H of the attenuation a(w): :$K(iw)\; =\; H(a(w))+\; i\; a(w)\; \backslash quad\; (1.4)$ For our calculations to be valid we must make two assumptions: (a) the absorption coefficient a(w) is strictly linear in the frequency, over the range of measurement. We can call it bw. (b)The wave motion is linear i.e. the principle of superposition is valid. The second assumption is of more fundamental nature, and give us a possibility to express any pulse U as a superposition of plane waves.〔 The reason why Futterman introduced the Hilbert transform was to impose causality on the wave pulse. Substituting this complex-valued wavenumber K(w) into solution (1.2) produces the following expression: :$U(r+\backslash bigtriangleup\; r,w)=U(r,w)\backslash exp((iH(bw))\backslash bigtriangleup\; r\; )\; \backslash quad\; (1.5)$ For a solution consistent with Futterman we can replace U(r,w) with the expression: :$U(r,w)\; =\; U\_0\; exp(iwt)\; \backslash quad\; (1.6)$ We can replace the distance increment ∆r by traveltime increment ∆t = ∆r/c. Here c is a constant. We can replace U0 with U' for correct scaling and equation (1.5) is expressed as :$U(t+\backslash bigtriangleup\; t,w)=U\text{'}\; exp(iw\backslash bigtriangleup\; t)\backslash exp((iH(bw))\backslash bigtriangleup\; t)\; \backslash quad\; (1.7)$ The sum of these plane waves gives the time-domain seismic signal, :$U(t+\backslash bigtriangleup\; t)\; =\backslash int\_0^\backslash infty\; U(t+\backslash bigtriangleup\; t,w)dw.\backslash quad\; (1.8)$ 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Dispersive body waves」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.